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More Transformations of Sine and Cosine - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

In the equation y=Asin(B(x-h)), A modifies the amplitude and B modifies the period; see sine and cosine transformations. The constant h does not change the amplitude or period (the shape) of the graph. It shifts the graph left (if h is negative) or right (if h is positive) and in the amount equal to h. The amount of horizontal shift is called the phase shift, which equals h.

I'm graphing transformations of sine and cosine and I've already talked about graphing y equals a sine of bx. So I want to talk about graphing y equals a sine of b times the quantity x minus h so this part is new. And before I get started with examples let's take a look at a demonstration that allows us to see what that x minus h does. Okay here we are looking at geometries sketch pad and I've got my equation g of x equals a times the sine of b times the quantity x minus h. I have sliders over here that allow me to control the values of a, b and h and here's my graphs of first of all the sine graph and second of all the transformed sine graph. Now right now a is 3 and b is 0.5 I'm going to leave those fixed for the moment. Let's adjust the value of h and see what happens, if I adjust it in the positive direction you can see that the shape of the graph doesn't change at all, but it shifts to the right.
It shifts to the right exactly as much as the value of h notice that the point, this point started at the origin but now it's a pi h let me show you again. Back at the origin and then up to the pi. So the h value if the h is pi that means this will shift to the right per unit. And what happens if h is negative it just shifts to the left. And so basically the value of h tells you what the horizontal shift is h equals negative 0.5 pi, negative one half pi. If you look at the formula you have g of x equals 3 sine 0.5 times x plus 0.5 pi that's x minus negative 0.5 pi. That means a shift to the left half of pi, so again if h is positive we shift to the right by that amount in this case 0.75 pi. If h is negative, we shift to the left by that amount.
Okay let's review what we just learned, the value of h controls the horizontal shift of the graph and if h is half of pi then we shift to the right half of pi. And when we're talking about sine and cosine graphs this is called phase shift. And the phase shift is exactly the horizontal shift of a sine or cosine graph. And it exactly equals h, so you don't have to go through a fancy formula to find the phase shift. Just write the sine or cosine function in this form and identify h that's the phase shift and we'll be using this when we're transforming sine and cosine graphs.

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